Induction proof...I'm stuck!

Hopefully someone can help me with this..I'm nearly done.

I'm in the induction step of the proof & here's what I have:

"So suppose that P(n) holds for n an integer and n>=5, we have that
2^n > n^2.
We wish to show that P(n+1) holds, that is that
2^(n+1) > (n+1)^2. This can be rewritten as
2(2^n) >n^2 +2n +1.
So, by our induction hypothesis and our Lemma (stated & proved earlier before this proof) we have
2(2^n) > 2^n >n^2 > 2n+1."

....This is where I'm stuck...is there a property that allows me to then just state "Thus 2(2^n) > n^2 +2n +1"?
Thanx for any feedback you provide!

Are you using the fact that 2^(n+1) = 2^n + 2^n ? I started to use this in my proof then realized that although this is true for a base 2 it is not true for a base > 2 (i.e. 3^(n+1) does not = 3^n + 3^n...
3^(n+1)=3(3^n).